Tag: coverage

Meta visualization

In this post we contrast the meta distribution of the SIR with the standard SIR distribution. The model is the standard downlink Poisson cellular network with Rayleigh fading and path loss exponent 4. The base station density is 1, and the users form a square lattice of density 5. Hence there are 5 users per cell on average.

Fig. 1: The setup. Base stations are red crosses, and users are blue circles. They are served by the nearest base station. Cell boundaries are dashed.

We assume the base stations transmit to the users in their cell at a rate such that the messages can be decoded if an SIR of θ =-3 dB is achieved. If the user succeeds in decoding, it is marked with a green square, otherwise red. We let this process run over many time slots, as shown next.

Fig. 2: Transmission success (green) and failure (red) at each user over 100 time slots.

The SIR meta distribution captures the per-user statistics, obtained by averaging over the fading, i.e., over time. In the next figure, the per-user reliabilities are illustrated using a color map from pure red (0% success) to pure green (100% success).

Fig. 3: Per-user reliabilities, obtained by averaging over fading. These are captured by the SIR meta distribution. Near the top left is a user that almost never succeeds since it is equidistant to three base stations.

The SIR meta distribution provides the fractions of users that are above (or below) a certain reliability threshold. For instance, the fraction of users that are at least dark green in the above figure.

In contrast, the standard SIR distribution, sometimes misleadingly called “coverage probability”, is just a single number, namely the average reliability, which is close to 70% in this scenario (see, e.g., Eqn. (1) in this post). Since it is obtained by averaging concurrently over fading and point process, the underlying network structure is lost, and the only information obtained is the average of the colors in Fig. 3:

Fig. 4: The standard SIR distribution only reveals the overall reliability of 70%.

The 70% reliability is that of the typical user (or typical location), which does not correspond to any user in our network realization. Instead, it is an abstract user whose statistics correspond to the average of all users.

Acknowledgment: The help of my Ph.D. student Xinyun Wang in writing the Matlab program for this post is greatly appreciated.

What to expect (over)

In performance analyses of wireless networks, we frequently encounter expectations of the form

\displaystyle\mathbb{E}\log(1+{\rm SIR}),\qquad\qquad\qquad\qquad\qquad\qquad (*)

called average (ergodic) spectral efficiency (SE) or mean normalized rate or similar, in units of nats/s/Hz. For networks models with uncertainty, its evaluation requires the use stochastic geometry. Sometimes the metric is also normalized per area and called area spectral efficiency. The SIR is expressed in the form

\displaystyle {\rm SIR}=\frac{h_y \|y\|^{-\alpha}}{\sum_{x\in\Phi} h_x \|x\|^{-\alpha}},

with Φ being the point process of interferers.
There are several underlying assumption made when claiming that (*) is a relevant metric:

  • It is assumed that codewords are long enough and arranged in a way (interspersed in time, frequency, or across antennas) such that fading is effectively averaged out. This is reasonable for several current networks.
  • It is assumed that desired signal and interference amplitudes are Gaussian. This is sensible since if a decoder is intended for Gaussian interference, then the SE is as if the interference amplitude were indeed Gaussian, regardless of its actual distribution.
  • Most importantly and questionably, taking the expectation over all fading random variables hx implies that the receiver has knowledge of all of them. Gifting the receiver with all the information of the channels from all interferers is unrealistic and thus, not surprisingly, leads to (*) being a loose upper bound on what is actually achievable.

So what is a more realistic and accurate approach? It turns out that if the fading in the interferers’ channels is ignored, i.e., by considering

\displaystyle {\rm SIR}^\prime=\frac{h_y \|y\|^{-\alpha}}{\sum_{x\in\Phi} \|x\|^{-\alpha}},

we can obtain a tight lower bound on the SE instead of a loose upper bound. A second key advantage is that this formulation permits a separation of temporal and spatial scales, in the sense of the meta distribution. We can write

\displaystyle {\rm SIR}^\prime=h_y\rho,\qquad\text{where }\; \rho=\frac{\|y\|^{-\alpha}}{\sum_{x\in\Phi} \|x\|^{-\alpha}}

is a purely geometric quantity that is fixed over time and cleanly separated from the time-varying fading term hy. Averaging locally (over the fading), the SE follows as

\displaystyle C(\rho)=\mathbb{E}_h \log(1+h\rho),

which is a function of (conditioned on) the point process. For instance, with Rayleigh fading,

\displaystyle C(\rho)=e^{1/\rho} {\rm Ei}_1(1/\rho),

where Ei1 is an exponential integral. The next step is to find the distribution of ρ to calculate the spatial distribution of the SE – which would not be possible from (*) since it is an “overall average” that lumps all randomness together. In the case of Poisson cellular networks with nearest-base station association and path loss exponent 2/δ, a good approximation is

Here s* is given by

\displaystyle s^{*\delta}\gamma(-\delta,s^*)=0,

and γ is the lower incomplete gamma function. This approach lends itself to extensions to MIMO. It turns out that the resulting distribution of the SE is approximately lognormal, as illustrated in Fig. 1.

Figure 1: Spatial distribution of ergodic spectral efficiency for 2×2 MIMO in a Poisson cellular network and its lognormal approximation (red).

For SISO and δ=1/2 (a path loss exponent of 4), this (approximative) analysis shows that the SE achieved in 99% of the network is 0.22 bits/s/Hz, while a (tedious) simulation gives 0.24 bits/s/Hz. Generally, for small ξ, 1/ln(1/ξ) is achieved by a fraction 1-ξ of the network. As expected from the discussion above, this is a good lower bound.

In contrast, using the SIR distribution directly (and disregarding the separation of temporal and spatial scales), from

\displaystyle \bar F_{\rm SIR}(\theta)=0.99 \quad\Longrightarrow\quad \theta=-20\text{ dB},

we would obtain an SE of only log2(1.01)=0.014 bits/s/Hz for 99% “coverage”, which is off by a factor of 16! So it is important that coverage be gleaned from the ergodic SE rather than a quantity subject to the small-scale variations of fading. See also this post.

The take-aways for the ergodic spectral efficiency are:

  • Avoid mixing time and spatial scales by expecting first over the fading and separately over the point process; this way, the spatial distribution of the SE can be obtained, instead of merely its average.
  • Avoid gifting the receiver with information it cannot possibly have; this way, tight lower bounds can be obtained instead of loose upper bounds.

The details can be found here.